Theory of the upper critical field in antiferromagnetic superconductors

We compute the temperature $T$ dependence of the upper critical field $H_{c2\mathrm{}}\mathrm{}\left(T\right)\mathrm{}$ in antiferromagnetic (AF) superconductors. Using a strong-coupling formalism we explicitly treat the effects of the molecular field $H_Q$, inelastic and elastic spin-fluctuation scattering and magnetic as well as nonmagnetic impurities. A sum rule is used to relate the $T$ dependence of $H_Q$ to that of the spin-fluctuation scattering. The decreased pair breaking observed below the Néel temperature in Sm${\mathrm{Rh}}_4$ ${\mathrm{B}}_4$ and the increased pair breaking seen in the AF Chevrel compounds will both occur in our theory for a reasonable choice of parameters. For larger values of the dimensionless spin-exchange coupling constant $N\left(0\right)\mathrm{}{J}^{\mathrm{cf}}$, spin-fluctuation-scattering effects dominate over those of $H_Q$ and decreased pair breaking is observed below $T_N$. For smaller values of the coupling constant, the converse is true. Impurity scattering is treated in a self-consistent fashion. As a consequence, the molecular field $H_Q$ is altered by nonmagnetic impurities. This leads to important pair-breaking effects in $H_{c2\mathrm{}}$. A physical manifestation of this pair breaking is a qualitative change in the shape of the $H_{c2\mathrm{}}$ versus $T$ curve, as nonmagnetic impurities are added. We give detailed predictions for the expected effects of these impurities on $H_{c2\mathrm{}}$ which can be tested experimentally.